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Exponential test-function characterization of Brownian motion

Let

W

be continuous with

W_{0} = 0

. If

e^{c W_{t}} - \frac{c ^{2}}{2} \int_{0}^{t} e^{c W_{s}} d s

is a martingale for every real

c

, then conditional expectations solve

E [e^{c W_{t}} ∣ F_{s}] = e^{c W_{s} + c^{2} (t - s) /2} .

Thus

W_{t} - W_{s}

is independent of

F_{s}

and distributed as

N (0, t - s)

, so

W

is Brownian motion.

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